11 Cylindrical and Spherical Coordinates - Handout

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Math 55 chapter 11

Transcript of 11 Cylindrical and Spherical Coordinates - Handout

  • Triple Integrals in Cylindrical& Spherical Coordinates

    Math 55 - Elementary Analysis III

    Institute of MathematicsUniversity of the Philippines

    Diliman

    Math 55 Cylindrical & Spherical Coordinates 1/ 23

  • Cylindrical Coordinates

    Consider a point P in space. Let Q be the projection of P ontothe xy-plane. Let r be the distance of Q from the origin, and be the angle between the positive x-axis and the line segmentOQ.

    Qr

    x

    y

    z

    P

    O

    The ordered triple (r, , z) is called the cylindrical coordinatesof P .

    Math 55 Cylindrical & Spherical Coordinates 2/ 23

  • Surfaces in Cylindrical Coordinates

    If r = c, a constant then we have a cylinder with radius |c|.

    Math 55 Cylindrical & Spherical Coordinates 3/ 23

  • Surfaces in Cylindrical Coordinates

    If = c, a constant then we have a half-plane.

    Math 55 Cylindrical & Spherical Coordinates 4/ 23

  • Surfaces in Cylindrical Coordinates

    If z = r, then we have a cone.

    Math 55 Cylindrical & Spherical Coordinates 5/ 23

  • Cartesian Cylindrical

    r

    x

    y

    z

    P

    O

    The relationship between the Cartesian coordinates P (x, y, z)and cylindrical coordinates (r, , z) is given by

    x = r cos

    y = r sin

    z = zMath 55 Cylindrical & Spherical Coordinates 6/ 23

  • Cartesian Cylindrical

    Remark

    1 By the distance formula, r2 = x2 + y2.

    2 dV = dz dA = rdz dr d

    Math 55 Cylindrical & Spherical Coordinates 7/ 23

  • Exercises

    1 Write the following equations in cylindrical coordinates.

    a. z =

    3x2 +

    3y2

    b. x2 + y2 = 2y

    c. z2 = 1 + x2 + y2

    d. 2x2 + 2y2 + z2 = 4

    2 A solid E lies within the cylinder x2 + y2 = 1, below theplane z = 4, and above the paraboloid z = 1 x2 y2.Find the volume of the solid E.

    3 Evaluate

    22

    4y24y2

    2x2+y2

    xz dz dx dy

    Math 55 Cylindrical & Spherical Coordinates 8/ 23

  • Spherical Coordinates

    Consider a point P in space. Let be the distance of P fromthe origin, be the same angle as in cylindrical coordinates, and be the angle between the positive z-axis and the line segment

    OP .

    x

    y

    z

    P

    O

    The ordered triple (, , ) is called the spherical coordinates ofP .

    Math 55 Cylindrical & Spherical Coordinates 9/ 23

  • Surfaces in Spherical Coordinates

    If = c, a constant then we have a sphere.

    Math 55 Cylindrical & Spherical Coordinates 10/ 23

  • Surfaces in Spherical Coordinates

    If = c, a constant then we have a half-plane.

    Math 55 Cylindrical & Spherical Coordinates 11/ 23

  • Surfaces in Spherical Coordinates

    If = c, a constant then we have a cone.

    Figure: 0 c pi2 Figure: pi2 c pi

    Math 55 Cylindrical & Spherical Coordinates 12/ 23

  • Spherical CartesianSuppose P (x, y, z) has Spherical coordinates (, , ).

    z

    rx

    y

    P

    Then z = cos. Let r be as in cylindrical coordinates. Thenr = sin. Therefore, x = r cos = sin cos andy = r sin = sin sin

    Math 55 Cylindrical & Spherical Coordinates 13/ 23

  • Spherical Cartesian

    The relationship between the Cartesian coordinates P (x, y, z)and spherical coordinates (, , ) is given by

    x = sin cos

    y = sin sin

    z = cos.

    Also, the distance formula shows

    2 = x2 + y2 + z2.

    Note that > 0 and 0 pi.

    Math 55 Cylindrical & Spherical Coordinates 14/ 23

  • Spherical CartesianExample

    Find the Cartesian coordinates of the point with sphericalcoordinates

    (2, pi4 ,

    pi3

    ).

    Solution.

    x = sin cos = 2(

    sinpi

    3

    ) (cos

    pi

    4

    )= 2

    (3

    2

    ) (2

    2

    )=

    6

    2

    y = sin sin = 2(

    sinpi

    3

    )(sin

    pi

    4

    )= 2

    (3

    2

    )(2

    2

    )=

    6

    2

    z = cos = 2(

    cospi

    3

    )= 2

    (1

    2

    )= 1.

    Math 55 Cylindrical & Spherical Coordinates 15/ 23

  • Spherical Cartesian

    The point with spherical coordinates(2, pi4 ,

    pi3

    )is the point with

    Cartesian coordinates (62 ,62 , 1).

    Math 55 Cylindrical & Spherical Coordinates 16/ 23

  • Spherical CartesianExample

    A point P has Cartesian coordinates (0, 2

    3,2). Find thespherical coordinates of P .

    Solution. =

    x2 + y2 + z2

    =

    0 + 12 + 4 =

    16 = 4

    cos =z

    =24

    = 12

    =2pi

    3,4pi

    3

    2pi

    3

    cos =x

    sin= 0

    =pi

    2,3pi

    2

    pi

    2because 0 pi since y > 0 Therefore, the sphericalcoordinates of P are (4, pi2 ,

    2pi3 ).

    Math 55 Cylindrical & Spherical Coordinates 17/ 23

  • Triple Integrals in Spherical Coordinates

    Suppose E is a solid given in spherical coordinates by

    E = {(, , )|a b, , c d}

    where a 0, 2pi and d c pi. We call this aspherical wedge.

    We subdivide E into smaller sub-wedges Eijk by means ofequally spaced spheres = i, half-planes = j and half-cones = k.

    Math 55 Cylindrical & Spherical Coordinates 18/ 23

  • Triple Integrals in Spherical Coordinates

    Consider the sub-wedge Eijk.

    The volume of thesub-wedge can beapproximated by arectangular box withdimensions

    , i, i sink

    If Vijk is the volume ofEijk, then

    Vijk 2i sink

    Math 55 Cylindrical & Spherical Coordinates 19/ 23

  • Triple Integrals in Spherical Coordinates

    Suppose (i, j , k) has Cartesian coordinates (xijk, y

    ijk, z

    ijk).

    ThenE

    f(x, y, z)dV = liml,m,n

    li=1

    mj=1

    nk=1

    f(xijk, yijk, z

    ijk)Vijk

    which gives the formulaE

    f(x, y, z)dV =

    dc

    ba

    f( sin cos , sin sin , cos)2 sinddd

    where E is the spherical wedge given by

    E = {(, , )|a b, , c d}

    Math 55 Cylindrical & Spherical Coordinates 20/ 23

  • Triple Integrals in Spherical Coordinates

    Example

    Evaluate

    E

    z dV where E lies between the spheres

    x2 + y2 + z2 = 1 and x2 + y2 + z2 = 4 in the first octant.

    Solution. The equations of the spheres in sphericalcoordinates are = 1 and = 2. Since E is in the first octant,

    E ={

    (, , ) : 1 2, 0 pi2, 0 pi

    2

    }.

    Hence,E

    z dV =

    pi2

    0

    pi2

    0

    21

    ( cos)2 sinddd

    =

    pi2

    0

    pi2

    0

    213 cos sinddd =

    15pi

    16

    Math 55 Cylindrical & Spherical Coordinates 21/ 23

  • Exercises

    1 Write the following equations in spherical coordinates.

    a. x = yb. z2 = x2 + y2

    c. x2 + z2 = 9d. x2 + y2 + z2 = 2z

    2 Evaluate

    B

    e(x2+y2+z2)

    32 dV where B is the unit ball

    centered at the origin.

    3 Use spherical coordinates to find the volume of the solidthat lies above the cone z =

    x2 + y2 and below the

    sphere x2 + y2 + z2 = z.

    4 Evaluate

    30

    9y20

    18x2y2x2+y2

    (x2 + y2 + z2) dz dx dy.

    5 Find the mass of a solid E with constant density if E liesabove the cone z =

    x2 + y2 and below the sphere

    x2 + y2 + z2 = 1.

    Math 55 Cylindrical & Spherical Coordinates 22/ 23

  • References

    1 Stewart, J., Calculus, Early Transcendentals, 6 ed., ThomsonBrooks/Cole, 2008

    2 Dawkins, P., Calculus 3, online notes available atwww.lamar.com/pauldawkins

    Math 55 Cylindrical & Spherical Coordinates 23/ 23